This question is somewhat related to this post.
Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$.
Assume that this current is passed through a system, which integrates for a finite time duration ($T_{int}$) on a capacitor $C$. One can then write an equation for the power of the noise voltage (integration of current produces voltage $V=\frac{1}{C}\int_{0}^{T}i(t)dt $ ) produced from this integration process ($\sigma_v^2$):
$$\begin{align}\sigma_{v}^{2} &=E\bigg(\frac{1}{C}\int_{0}^{T_{int}}i_{n}(t_{1})dt_{1}\frac{1}{C}\int_{0}^{T_{int}}i_{n}(t_{2})dt_{2}\bigg) \\&=\frac{1}{C^{2}}\int_{0}^{T_{int}}\int_{0}^{T_{int}}E(i_{n}(t_{1})i_{n}(t_{2}))dt_{1}dt_{2} \\&=\frac{1}{C^{2}}\int_{0}^{T_{int}}\int_{0}^{T_{int}}\sigma_i^2\delta(t_{2}-t_{1})dt_{1}dt_{2} \\&=\frac{1}{C^{2}}\int_{0}^{T_{int}}\sigma_i^2dt_{2} =\frac{1}{C^{2}}\sigma_i^2\cdot T_{int}\end{align}$$
Here $\delta$ denotes the dirac delta impulse function.
This is a similar result obtained to what was obtained in the post linked above. The problem with the above equation is that it is dimensionally inconsistent (there should be a $T_{int}^2$ factor in the numerator to make it consistent.)
Can someone kindly point out the error in the above argument?